What is the most Beautiful Equation?

What makes an equation “beautiful?” It is kind of complicated. It is a mix of lizard brain and analytical brain: it has to be profound and elegant (simple) at the same time. Oh yeah and there has to be a single equality sign somewhere.

Let me give you some examples.

Number one:

x = x.

Wow it is simple but it just says a thing is equal to itself. It is obvious to anyone and doesn’t communicate anything interesting. It looks cool though as it is symmetrical.

Number two:

1+1 = 0.

Whoa? No that should be two not zero! Well not if you consider a field with two elements labelled “0” and “1.” We are not talking about running through corn fields hand in hand on a first date. A “field” is some mathematical construct that defines addition and multiplication in an abstract way. This is mathematical snobbery. Only the ones (pun intended) in the know will say “cute.” On the other hand the low life uneducated proles will say “nonsense.” I am being sarcastic of course. More of an inside joke. Not really profound then.

Number three:

a^2 + b^2 = c^2

If you are like me you will cringe seeing this equation. It reminds me of the dark ages when I hated math. You had to learn formulas by rote and we would write them on our cheat sheets, on our underwear or sleeves: anything really to pass the exam. But it actually has a beautiful proof I found out later. See for yourself.

Hint: relation to boring math: (a+b)x(a+b) – cxc = 2xaxb, so axa+bxb+2xaxb – cxc = 2xaxb, so axa+bxb=cxc. The formula is beautiful and the geometrical proof is beautiful. Hey this might be a good candidate for a hunk equation.

Number four:

V – E + F = X.

What? That seems like a nonsensical equation. What is V? What is E? What is F and what is X? Hold on. This is one of the many beautiful concoctions cooked up by Leonhard Euler. It explains why you cannot wrap a piece of paper on a sphere without cutting it. Try this at home. In plain language it says:

The number of vertices minus the number of edges plus the number of faces is equal to the Euler characteristic.

Visually for these approximations of the sphere the sum on the left is always equal to two.

Do the boring math for these examples and see for yourself that the formula works. Try it for a soccer ball as well. Of course this is not a proof. No worries Euler has one.

For planar surfaces like a piece of paper the sum on the left is always equal to one.

We have a simple formula, cool math, a simple proof and it is very useful. This is definitely a babe equation. What are the other shapes that correspond to other Euler characteristics? Welcome to the fun world of mathematical topology. Hint: a donut has an Euler characteristic equal to zero.

Number five:

e^(i pi) + 1 = 0.

To me this is the pièce de résistance. It is the crown jewel of Euler’s work. Why do I think it is so cool? It combines five fundamental numbers that are all over mathematics in a single equation. These numbers are:

0, 1, e, i and pi.

But in a nutshell what this equation states is that if you rotate a vector of unit length by 180 degrees you will get a new vector that points in the opposite direction. And if you add the start vector with the rotated one, they will cancel each other out.

This is my all-time favorite equation.

There are of course many others. And please feel free to add your favorites in the comments below.

But before I let you go…

Physics on the other hand mostly comes up with ugly equations. Poor folks they actually have to match their equations to real experiments. Mathematicians on the other hand have free reign. The big mystery to me is why sometimes mathematics is helpful in physics.

Here is the Lagrangian of the standard model that explains most interactions between particles. Try to spot the term that corresponds to the Higgs Boson, also known as the “God Particle.”

Ok Ok you might ask where is the equality sign? I should have added L = {monstrous mess depicted in the jpg above}.